Wojciech Szczesny has suggested that Arsenal's home victory over Manchester United was underpinned by a sense of liberation following the virtual collapse of their Premier League title hopes.

The 1-0 win on Sunday was a welcome tonic in the wake of the team's recent slump, which has seen them drift from contenders to rank outsiders, but it also raised the question as to how they could beat the league leaders yet slip up at home to other teams.

Szczesny admitted that pressure has been a factor. Arsenal were under suffocating amounts of it as they toiled through March and April, with their championship hopes all but extinguished by the defeat at Bolton Wanderers on the Sunday before last.

Against United, though, there seemed to be greater freedom and enjoyment about their football and the Polish goalkeeper accepted that the challenge now is to reproduce that form when it matters the most. The 21-year-old also had no doubts that his team-mates would be able to do so next season.

"We played with the handbrake off [against United] because there was less pressure on us," Szczesny said. "We really enjoyed our game and that made a difference. Also, we defended very, very solidly at the back, which wasn't the case in the last few weeks."

It was put to Szczesny that Arsenal could not rely on playing without pressure situations if they want to win trophies. "Of course," he replied. "That's why next season we have to prove we can step up. We believe we are good enough to make that extra step to win the league."

Arsenal have now beaten Chelsea, Barcelona and United at home this season but have dropped points against a clutch of lower-ranked teams. Prior to United's visit they had drawn the previous three home league fixtures – against Sunderland, Blackburn Rovers and Liverpool.

The Arsenal manager, Arsène Wenger, has complained that those teams and plenty of others had "only defended", which led to his side's frustration, yet it is incumbent on the Gunners to break down such opponents when the pressure is on. Wenger has conceded that his team "maybe lack a bit of maturity under pressure".

"We all take individual responsibilities for our mistakes," Szczesny added, "and everyone has played their part in this season's disappointments. We just have to take it as a team. You can't blame the boss because he stood by the team all season and supported us.

"There's a lot of frustration because we work hard and do our best, and it's frustrating not to be on the side that wins the championship. But we have showed we have the quality and now it's a matter of can we step up next season?

"Manchester United have been more consistent and they didn't drop silly points like we did in the last couple of weeks. Realistically the championship chances were already over before the United game, but we wanted to play for pride and show we are good enough to challenge them."

Wenger's starting line-up for the game – average age 23 years 296 days – was the youngest to play for anyone in the Premier League this season, and there was encouragement in how Aaron Ramsey stepped in for the injured Cesc Fábregas to pair up with Jack Wilshere in front of Alex Song in a slightly remodelled midfield formation.

If it offered a glimpse as to how they might cope in the event of Fábregas leaving for Barcelona this summer, although that prospect is surely not one to be embraced as they would carry even less experience without their Catalan captain.

Bacary Sagna, the defender, said that he would understand it if Fábregas wanted to win trophies with another club, after nearly six seasons without one at Arsenal, but Szczesny disagreed. "I wouldn't understand leaving Arsenal," he said, "because I believe being at this club is the best thing in the world. So, no, I wouldn't understand leaving."

As for Wenger being pressed upstairs into an administrative role, or out of the club, Szczesny was derisive. "It's ridiculous," the Pole added. "He's the best manager in the world so I don't understand why anyone would want him to leave. Let's be patient and show next year we are good enough to win the league."